On the morning of March 1, 1917, virtually every American newspaper published a bombshell story: a report on a telegram from the German foreign secretary, Arthur Zimmermann, proposing an alliance with Mexico. He offered his country’s support to Mexico for reconquering “the lost territory in Texas, New Mexico, and Arizona” in exchange for a Mexican attack on the United States should the Americans enter the war on the side of the Allies.

World War I had entered a crucial state in the winter of 1916-17, with neither the Allies nor the Central Powers having the resources to push through to victory. To be sure, the Allies had more soldiers on the battlefield. Nonetheless, the German army remained deep in French and Russian territory, and deadly submarine warfare deprived millions of people of the substances of life. War-weary populaces on both sides yearned for an end of watching their youth fed through bloody meat grinders.

British diplomacy (as well as covert propaganda) sought to persuade an obstinate President Wilson to reverse his 1916 campaign promise of neutrality and enter the war on the side of the Allies. German propagandists sought to reinforce the American public’s abhorrence of the war.

Varying accounts of just how the famed “Zimmermann telegram” got into the hands of British signal intelligence, and then to Wilson’s White House, have been published over the years, most notably by historian Barbara Wertheim Tuchman. But she wrote without access to German foreign office papers, leading one German historian to comment that her work was plagued by “errors, inaccuracies and obsolete theses.”

Now comes what should be the definitive work on the subject. The German-born Thomas Boghardt is senior historian for the U.S. Army Center of Military History in Washington. He obtained access to German diplomatic archives that enabled him to dash some prevailing myths and to give a very readable account of the episode.

Foremost is his debunking of the notion that the “Mexican plot” was part of a carefully planned German strategy to gain a foothold in the Western Hemisphere. In fact, the scheme originated with a minor foreign office official, Hans Arthur von Kemnitz, who wrote the first draft of the telegram. Zimmermann spent “hardly any time” studying it before signing it — likely because he was preoccupied with drafting a new German policy announcing “unrestricted submarine warfare” against shipping to England. When another functionary saw news reports of the telegram’s disclosure, he exclaimed, “Kemnitz, that fantastic idiot, has done this!”

So how did the telegram end up in the hands of the British, who were quite happy to pass it to the United States to illustrate German perfidy? Here is where Mr. Boghardt does a superb job of stripping away a veil of misinformation that Britain draped over the story.

Early in the war, the British Royal Navy severed Germany’s trans-Atlantic submarine cables. (A young Winston Churchill, first lord of the Admiralty, ordered the creation of a section to handle the decoded intercepts, and “Room 40” became vital to Allied victory in the war.) Then came a windfall for the Britons.

Eager to play the role of peacemaker by communicating with both warring parties, Wilson’s close aide, Col. Edward House, had the State Department allow the transmission of enciphered German messages in its diplomatic cables among Washington, London, Copenhagen and Berlin.

Mr. Boghardt writes, “Trusting in the Germans’ good faith, the Americans accepted numerous encrypted messages without being knowledgeable of their actual content.” Berlin officials would hand a message to the U.S. ambassador in Berlin, James Gerard, who would transmit it to the American legation in Copenhagen, which would forward it via the London embassy to the State Department in Washington and then to the German ambassador. The traffic flowed in both directions.

All the while, Room 40 read all this traffic, unbeknownst to the United States — German messages along with American cables. Thus, the intercept of the Zimmermann telegram, with its explosive offer to Mexico.

The British code titan, Navy Capt. William Reginald “Blinker” Hall, realized that the information could bring the United States into the war on the Allied side. He “feared that the Americans might discover Room 40’s eavesdropping on their communications. If the Americans had made an educated guess, the telegram would not only have caused a diplomatic scandal between Washington and London, but the State Department would probably have changed its codes, depriving Room 40 of an important intelligence source.”

So Hall concocted a cover story — a claim that the British obtained the telegram when it was forwarded by the German Embassy in Washington to Mexico City. (Indeed, the Britons did just that in due course, by bribing a Mexican telegraph employee.)

Then Blinker Hall made a bold move. When the Foreign Office dithered over sharing the information with Washington, Hall called in a U.S. intelligence officer in London and gave him the telegram. As Mr. Boghardt writes, “In other words, the director of naval intelligence had unilaterally made the decision to share a highly sensitive piece of information with a foreign power without proper authorization from his own government.”

In reality, the German proposal was daft. To be sure, the United States was weak militarily, but Mexico, emerging from years of bloody revolution, was in no condition to challenge its stronger neighbor. No matter. The Wilson White House released the telegram to the Associated Press, and headlines blazed news of the “threat.”

Historians for the most part assert that the Zimmermann telegram drove Wilson’s decision to enter the war a few days later. But Mr. Boghardt did an intense analysis of the editorial position of leading U.S. papers and found that few changed their stances toward the war because of the telegram. The key precipitant toward war was the German sinking of a British liner, killing two U.S. women, and the announcement of unrestricted submarine warfare.

Nonetheless, Mr. Boghardt concludes, “The United States certainly would have entered World War I regardless of the telegram, but by removing Wilson’s final doubts about the wisdom of joining the Allies, it accelerated U.S. intervention, though perhaps only by a few weeks.”

Mr. Boghardt’s work is a masterpiece of intelligence writing. By following the hard evidence rather than relying on historical assumptions, he provides an incisive case study on how intelligence can affect national affairs.